#496664

order	512
nilpotency class	3
isoclinism	isoclinic class of #3198
automorphism	?
homology	Abelianization (H₁) = C₂³×C₈ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₈	C₂²×D₄
commutator	C₂×C₄	C₂³×C₈
Frattini, cosocle	C₂²×C₈	C₂⁴
socle	C₂²	C₂²×C₄⋊C₈
prime radical	C₂²×#41	C₂×C₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₈ ◁ C₂²×#131 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

728 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 40 points:

(1,2)(3,7)(4,6)(5,8)(9,10,14,20,15,19,29,35,16,21,28,18,13,30,12,27)(11,17,25,34,26,33,37,40,23,32,38,31,24,39,22,36);(1,3)(2,5)(4,8)(6,7)(9,11)(10,17)(12,22)(13,24)(14,25)(15,26)(16,23)(18,31)(19,33)(20,34)(21,32)(27,36)(28,38)(29,37)(30,39)(35,40);(3,8)(5,7)(9,12,13,28,16,29,15,14)(10,18,19,27,21,20,30,35)(11,22,24,38,23,37,26,25)(17,31,33,36,32,34,39,40);(2,6)(5,7)(11,23)(17,32)(22,37)(24,26)(25,38)(31,34)(33,39)(36,40)

Polycyclic presentation:

2^9|a^2=g,c^2=f,f^2=i,g^2=h,h^2=i,b^a=be,c^a=cf,d^a=de,f^a=fi,c^b=ce,d^b=di

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 403860);